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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mexval | Structured version Visualization version GIF version | ||
| Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.) | 
| Ref | Expression | 
|---|---|
| mexval.k | ⊢ 𝐾 = (mTC‘𝑇) | 
| mexval.e | ⊢ 𝐸 = (mEx‘𝑇) | 
| mexval.r | ⊢ 𝑅 = (mREx‘𝑇) | 
| Ref | Expression | 
|---|---|
| mexval | ⊢ 𝐸 = (𝐾 × 𝑅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mexval.e | . 2 ⊢ 𝐸 = (mEx‘𝑇) | |
| 2 | fveq2 6906 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇)) | |
| 3 | mexval.k | . . . . . 6 ⊢ 𝐾 = (mTC‘𝑇) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾) | 
| 5 | fveq2 6906 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇)) | |
| 6 | mexval.r | . . . . . 6 ⊢ 𝑅 = (mREx‘𝑇) | |
| 7 | 5, 6 | eqtr4di 2795 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅) | 
| 8 | 4, 7 | xpeq12d 5716 | . . . 4 ⊢ (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅)) | 
| 9 | df-mex 35492 | . . . 4 ⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡))) | |
| 10 | fvex 6919 | . . . . 5 ⊢ (mTC‘𝑡) ∈ V | |
| 11 | fvex 6919 | . . . . 5 ⊢ (mREx‘𝑡) ∈ V | |
| 12 | 10, 11 | xpex 7773 | . . . 4 ⊢ ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V | 
| 13 | 8, 9, 12 | fvmpt3i 7021 | . . 3 ⊢ (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅)) | 
| 14 | xp0 6178 | . . . . 5 ⊢ (𝐾 × ∅) = ∅ | |
| 15 | 14 | eqcomi 2746 | . . . 4 ⊢ ∅ = (𝐾 × ∅) | 
| 16 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅) | |
| 17 | fvprc 6898 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mREx‘𝑇) = ∅) | |
| 18 | 6, 17 | eqtrid 2789 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅) | 
| 19 | 18 | xpeq2d 5715 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅)) | 
| 20 | 15, 16, 19 | 3eqtr4a 2803 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅)) | 
| 21 | 13, 20 | pm2.61i 182 | . 2 ⊢ (mEx‘𝑇) = (𝐾 × 𝑅) | 
| 22 | 1, 21 | eqtri 2765 | 1 ⊢ 𝐸 = (𝐾 × 𝑅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 × cxp 5683 ‘cfv 6561 mTCcmtc 35469 mRExcmrex 35471 mExcmex 35472 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-mex 35492 | 
| This theorem is referenced by: mexval2 35508 msubff 35535 msubco 35536 msubff1 35561 mvhf 35563 msubvrs 35565 | 
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